1. **State the problem:** Find the value of the determinant of the matrix $\begin{bmatrix} x & x & a \\ x & a & x \\ a & 2 & x \end{bmatrix}$. 2. **Recall the formula for a 3x3 determinant:** $$\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the matrix is $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$. 3. **Assign values:** $a = x, b = x, c = a$ $d = x, e = a, f = x$ $g = a, h = 2, i = x$ 4. **Calculate each minor:** - $ei - fh = a \cdot x - x \cdot 2 = ax - 2x = x(a - 2)$ - $di - fg = x \cdot x - x \cdot a = x^2 - ax = x(x - a)$ - $dh - eg = x \cdot 2 - a \cdot a = 2x - a^2$ 5. **Substitute into determinant formula:** $$\det = x \cdot x(a - 2) - x \cdot x(x - a) + a \cdot (2x - a^2)$$ 6. **Simplify each term:** - $x \cdot x(a - 2) = x^2(a - 2) = x^2a - 2x^2$ - $x \cdot x(x - a) = x^2(x - a) = x^3 - ax^2$ - $a \cdot (2x - a^2) = 2ax - a^3$ 7. **Put it all together:** $$\det = (x^2a - 2x^2) - (x^3 - ax^2) + (2ax - a^3)$$ 8. **Distribute the minus sign in the second term:** $$\det = x^2a - 2x^2 - x^3 + ax^2 + 2ax - a^3$$ 9. **Combine like terms:** - Combine $x^2a$ and $ax^2$ to get $2ax^2$ So, $$\det = -x^3 + 2ax^2 - 2x^2 + 2ax - a^3$$ **Final answer:** $$\boxed{\det = -x^3 + 2ax^2 - 2x^2 + 2ax - a^3}$$